Transcript Section 3.1 Graphs
Section 3.1
Graphs
OBJECTIVES
A Given an ordered pair of numbers, find its graph, and vice versa.
OBJECTIVES
B Graph lines by finding two or more points satisfying the equation of the line.
OBJECTIVES
C Graph lines by finding the x- and y- intercepts.
OBJECTIVES
D Graph horizontal and vertical lines.
DEFINITION
Standard Form of Linear Equations Ax + By = C
PROCEDURE
Finding the Intercepts Let y = 0, (x, 0) is x -intercept. Let x = 0, ( 0, y) is y -intercept.
RULE
Graphing Horizontal and Vertical Lines Y = C is a horizontal line.
X = C is a vertical line.
Chapter 3 Graphs and Functions Section 3.1A
Practice Test
Exercise #1
A
a .
Graph.
5 y
– 5 3 • 2 5 x
– 5
A
a .
Graph.
5 y
– 5 4 • –2
B
• 5 x – 5
A
a .
Graph.
5 y
– 5 –1 –2
C
• – 5 •
B
• 5 x
A
a .
Graph.
– 5
D
• 5 3
C
–1 • y
•
B
• 5 x – 5
A
a .
Graph.
D
• 5 y – 5
C
•
•
B
• 5 x – 5
b. Find the coordinates of the points in the figure.
A B
C
D
E B
5 y – 5
D A E
5 x
C
– 5
Chapter 3 Graphs and Functions Section 3.1A
Practice Test
Exercise #2
Graph the solutions to 3x – y = 3 .
If x = 0 0 – y = 3 y = – 3
0,–3
If y = 0 3x – 0 = 3 3x = 3 x = 1
Graph the solutions to 3x – y = 3 .
x
0
y
–3 y 1 0 • x • 3x – y = 3
Chapter 3 Graphs and Functions Section 3.1C
Practice Test
Exercise #3
Find the x- and y- intercepts of y = 3x + 2 then graph the solutions to the equation.
and x-intercept: Let y = 0 0 = 3x + 2 –2 = 3x 3
x
– intercept :
–2 3 ,0
Find the x- and y- intercepts of y = 3x + 2 then graph the solutions to the equation.
and y-intercept: Let x = 0 y = 3 0 + 2 y = 2
y
–
Find the x- and y- intercepts of y = 3x + 2 then graph the solutions to the equation.
5 y y = 3x + 2 and
0, 2
– 5 –2 3 , 0 5 x – 5
Chapter 3 Graphs and Functions Section 3.1D
Practice Test
Exercise #4
Graph the solutions to: a.
3x = – 9 b.
2y = – 4 5 y – 5 x = – 3 – 5 y = – 2 5 x
Section 3.2
Introduction to Functions
OBJECTIVES
A Find the domain and range of a relation.
OBJECTIVES
B Use the vertical line test to determine if a relation is a function.
OBJECTIVES
C Find the domain of a function defined by an equation.
OBJECTIVES
D Find the value of a function.
DEFINITION
Relation, Domain, and Range Relation: A set of ordered pairs.
DEFINITION
Relation, Domain, and Range Domain: A set of first coordinates.
DEFINITION
Relation, Domain, and Range Range: A set of second coordinates.
DEFINITION
Function A relation in which no two different ordered pairs have the same first coordinates.
PROCEDURE
Vertical Line Test If a vertical line intersects the graph more than once, the relation is not a function.
DEFINITION
Linear Function f (x) = mx + b
DEFINITION
Function 1. A function assigns exactly one range value to each domain value.
DEFINITION
Function 2. A function is a relation in which no two ordered pairs have the same first coordinate.
DEFINITION
Function 3. A function assigns one range to each domain.
Chapter 3 Graphs and Functions Section 3.2A
Practice Test
Exercise #5
Find the domain and range of the relation {(1, 3), (2, 5), (3, 7), (4, 9)}.
Domain = {1, 2, 3, 4} Range = {3, 5, 7, 9}
Chapter 3 Graphs and Functions Section 3.2A
Practice Test
Exercise #7c
Find the domain and range of the relation.
Domain
x
Ran ge
y
y
5 y – 5 – 5 y = – 4 – x 2 5 x
Chapter 3 Graphs and Functions Section 3.2B
Practice Test
Exercise #8b
Use the vertical line test to determine whether the graph of the given relation defines a function.
Not a function 4x 2 +
y
2 = 4 5 y – 5 5 x – 5
Chapter 3 Graphs and Functions Section 3.2C
Practice Test
Exercise #9
Find the domain of the function.
a.
y = 2 x + 3 Domain = {x | x is a real number ° – 3} b.
y = x + 2 Domai n
x x
– 2}
Chapter 3 Graphs and Functions Sections 3.2D
Practice Test
Exercise #10
Let ƒ(x) = 4x – 3. Find: ƒ = 8 – 3 = 5
Let ƒ(x) = 4x – 3. Find: = 5 ƒ = 4 – 3 = 1
Let ƒ(x) = 4x – 3. Find: = 5 = 1
= 4
Chapter 3 Graphs and Functions Section 3.2D
Practice Test
Exercise #11c
Let ƒ = {(1, 4), (2, -1), (3, 2)}. Find:
= –5
Section 3.3
Using Slopes to Graph Lines
OBJECTIVES
A Find the slope of a line passing through two given points.
OBJECTIVES
B Use the definition of slope to decide if two lines are perpendicular, or parallel.
OBJECTIVES
C Graph a line given its slope and a point on the line.
OBJECTIVES
D Find the slope and y intercept given the equation.
DEFINITION
Slope m =
y 2 x 2
– y
1
– x
1
RULES
Slopes of Parallel Lines Two lines with slopes m 1 and m 2 are parallel if m 1 = m 2 .
RULES
Slopes of Perpendicular Lines Two lines with slopes m 1 and m 2 are perpendicular if
m
1 • m 2 = – 1.
DEFINITION
The Slope-Intercept Form of a Line y = mx + b m is slope; b is y - intercept.
Chapter 3 Graphs and Functions Section 3.3B
Practice Test
Exercise #13b
A line L 1 has slope 3 2 . Determine whether the line through the two given points is parallel or perpendicular to L 1 .
A
and
m
=
= – 6 + 4 – 4 + 1 = –2 –3 = 2
L
1 2 AB is neither parallel nor perpendicular.
Chapter 3 Graphs and Functions Section 3.3B
Practice Test
Exercise #14
B
y
– 1 – 1 = 3 2 negative reciprocal
y
3
y + 2 = – 3 y = –5
Chapter 3 Graphs and Functions Section 3.3C
Practice Test
Exercise #15
A line has y-intercept 1 and has a slope – 1 2 .
Graph this line.
5 y – 5 5 x – 5
Chapter 3 Graphs and Functions Section 3.3D
Practice Test
Exercise #16
Find the slope and y –intercept of 4x – y = 3.
4x – y + y = 3 + y 4x – 3 = 3 – 3 + y y = 4x – 3 slope : 4 y-intercept : – 3
Section 3.4
Equations of Lines
OBJECTIVES
Find the equation and graph of a line given:
A Two points.
OBJECTIVES
Find the equation and graph of a line given:
B One point and the slope.
OBJECTIVES
Find the equation and graph of a line given:
C The slope and the y intercept.
OBJECTIVES
Find the equation and graph of a line given:
D One point and the fact that the line is parallel or perpendicular to a given line.
OBJECTIVES
Find the equation and graph of a line given:
E The slope is that of a horizontal or vertical line.
OBJECTIVES
Find the equation and graph of a line given:
F D Mathematical modeling application.
OBJECTIVES
Find the equation and graph of a line given:
G The graph of the line.
Point-Slope Form of a Line A line going through points (x (x 2 , y 2 ) is given by 1 , y 1 ) and 1 =
1
x
2
x
1
where m =
y
2 – y 1
x
2 – x 1
DEFINITION
The Slope-Intercept Form of a Line y = mx + b slope y-intercept
Chapter 3 Graphs and Functions Section 3.4A
Practice Test
Exercise #17
Find an equation of the line through (4, 3) and (2, 4). Then write the equation in standard form and graph the line.
y
1
x
1 –
y
–
x
2 2
2
y – 4 = 3 – 4 4 – 2
x – 2
y – 4 = –1
x – 2 2
Find an equation of the line through (4, 3) and (2, 4). Then write the equation in standard form and graph the line.
y – 4 =
–1
x – 2 2
2y – 8 = –x + 2 x + 2y – 8 = 2 x + 2y = 10
Find an equation of the line through (4, 3) and (2, 4). Then write the equation in standard form and graph the line.
5 y x + 2y = 10 – 5 5 x – 5
Chapter 3 Graphs and Functions Sections 3.4B
Practice Test
Exercise #18
Find an equation of the line with slope –2 and passing through the point (2, –3). Then graph the line.
1
y
y + 3 = –2x + 4 2x + y + 3 = 4 2x + y = 1
Find an equation of the line with slope –2 and passing through the point (2, –3). Then graph the line.
5 y – 5 2x + y = 1 5 x – 5
Chapter 3 Graphs and Functions Section 3.4C
Practice Test
Exercise #19
A line has slope 3 and y-intercept 2. Find the slope-intercept equation of this line and graph the line.
Given m = 3, b = 2 y = mx + b y = 3x + 2
A line has slope 3 and y-intercept 2. Find the slope-intercept equation of this line and graph the line.
5 y y = 3x + 2 – 5 5 x – 5
Chapter 3 Graphs and Functions Section 3.4D
Practice Test
Exercise #20
Find an equation of the line through the point (1,2) and: a.
Parallel to the line 2x – 3y = 5. Write in standard form.
2x – 3y = 5
–3y = –2x + 5 y = 2 3 x – 5 3
y
– 2 =
2 3
x
– 1
Find an equation of the line through the point (1,2) and: a.
Parallel to the line 2x – 3y = 5. Write in standard form.
3y – 6 = 2x – 2 –6 = 2x – 3y – 2 –4 = 2x – 3y 2x – 3y = –4
Find an equation of the line through the point (1,2) and: b.
Perpendicular to the line 2x – 3y = 5. Write in standard form.
y
– 2 = –
3 2
x
– 1
negative reciprocal 2y – 4 = –3x + 3 3x + 2y – 4 = 3 3x + 2y = 7
Chapter 3 Graphs and Functions Section 3.4E
Practice Test
Exercise #22
Given that a line passes through the point (– 2, 4), find the equation of the line if it is a vertical line.
y x = – 2 x
Chapter 3 Graphs and Functions Section 3.4F
Practice Test
Exercise #23
The points in the table represent a linear model. Plot the points on a graph, draw the line of best fit, and write the equation of that line in slope-intercept form.
X y
5 6 7 8 9 3 2 5 4 6 y x
Graph.
5 y 1 2 – 5 5 x – 5
Section 3.5
Linear Inequalities in Two Variables
OBJECTIVES
A Graph linear inequalities.
OBJECTIVES
B Graph inequalities involving absolute values.
DEFINITION
Linear Inequality In Two Variables or
C
A and B are not both zero.
PROCEDURE
Graphing a Linear Inequality 1. Graph boundary line. If the
line; otherwise, it is dashed.
PROCEDURE
Graphing a Linear Inequality 2. Choose a test point not on the line.
if possible.
PROCEDURE
Graphing a Linear Inequality 3. If the test point satisfies the inequality, shade the region containing the test point.
PROCEDURE
Graphing a Linear Inequality 3. Otherwise, shade the region on the other side of the line. The shaded region represents all solutions.
Chapter 3 Graphs and Functions Section 3.5A
Practice Test
Exercise #24
Graph.
a. x + 4y < 4 y x
Graph.
Related boundary: y = –1 y x
Chapter 3 Graphs and Functions Section 3.5B
Practice Test
Exercise #25a
x
Graph.
x
x = –6 y x = 4 x
Chapter 3 Graphs and Functions Additional
Exercises